What is the Four-Color Theorem?

Date: 04/03/98 at 10:44:59
From: Doctor Daniel
Subject: Re: Color Theorem
Hi there,
You asked what the "color theorem" is. I think you mean the Four
Color Theorem, although there may be other things you mean here.
Basically, here's what this Theorem states:
Suppose I give you any map that is on a flat piece of paper. (That
is, it doesn't look like a doughnut, or have 2 sides, or anything like
that. It's two-dimensional.)
Now, on that map, suppose that there are countries that are "simple,"
which means that no country has a part that's not connected to the
rest of the country. (So something like Alaska being part of the rest
of the U.S. isn't allowed.)
So if I give you one of these maps, with the boundaries between
countries drawn, the Four Color Theorem says that it's possible to
give each country a color and have no two countries that touch along
an edge with the same color.
I always think of those puzzles of the lower 48 states that I put
together when I was very small; the Theorem says essentially that no
matter what the states are shaped like, there's some way the pieces
can be in only 4 colors and when you are done there'll be no
adjacent pieces the same color.
The Theorem is also very important because it was the first major
theorem that was partially proven by a computer. Basically, the
authors used a computer to find roughly 1000 small maps and showed
that if the Theorem was false, one of them had to require 5 colors.
Then they made the computer color them with 4 colors, showing that
none required 5 colors. So the theorem is true.
A lot of mathematicians argued for quite a while about whether it's a
good proof, since it's somehow less beautiful than a lot of other math
out there. But most, at this point, accept that the theorem is
actually true.
Try making some maps of your own and seeing how to 4-color them. You
might also try seeing what you need to make sure it needs 4 colors and
not 3.
Good luck!
-Doctor Daniel, The Math Forum
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